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This is a drawing which shows how some comets approach the sun.
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Comet Trajectories

As comets pass into the inner solar system from the outer solar system, they stay in the solar system, in orbit around the sun, only if they are captured into an orbit. That is, the path they follow must change to an elliptic or circular path if they are to stay in the solar system.

This can happen through the action of the gravity of Jupiter or the Sun.

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